Suppose that λ 1 ( A ) ≥ λ 2 ( A ) ≥ ⋯ ≥ λ n ( A ) are (real) eigenvalues of a Hermitian matrix A and denote the empirical measure by L A := 1 n ∑ i = 1 n δ λ i ( A ) . Let F A to be distribution function related to the (counting) measure L A . We have the following integration by parts formula ∫ g d L A − ∫ g d L B = ∫ ( F A − F B ) d g ,in which g : R → R is a bounded function with bounded total variation. I am curious whether the integration by parts in this case is connected to Abel's summation by parts, and if so, can anyone explicitly present the connection?

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Suppose that       λ    1    (  A  )  ≥      λ    2    (  A  )  ≥  ⋯  ≥      λ    n    (  A  ) are (real) eigenvalues of a Hermitian matrix   A and denote the empirical measure by       L    A    :=      1    n        ∑          i      =      1        n        δ                  λ        i            (      A      )      . Let       F    A   to be distribution function related to the (counting) measure       L    A  . We have the following integration by parts formula  ∫  g        d        L    A    −  ∫  g        d        L    B    =  ∫  (      F    A    −      F    B    )        d    g  ,in which   g     :      R    →      R   is a bounded function with bounded total variation. I am curious whether the integration by parts in this case is connected to Abel&#039;s summation by parts, and if so, can anyone explicitly present the connection?

Jamiya Costa · Accepted Answer

Sure, just take   g to be a distribution function of some discrete probability distribution       L    g   (which is always bounded w/ bounded TV). Let   S be the union of all the supports of       L    g  ,       L    A  ,       L    B  , and let   I  =  [  n  ] be the index for the set   S, so that you can rewrite the LHS as:      ∑          i      ∈      [      n      ]        g  (      s    i    )  (      δ                  s        i            ∈              S        A              −      δ                  s        i            ∈              S        B              )and the right hand side is      ∑          i      ∈      [      n      ]        (  g  (      s    i    )  −  g  (      s          i      −      1        )  )  (      F    A    −      F    B    )  (      s    i    )This is basically summation by parts formula (where we note       δ                  s        i            ∈              S        A              =      F    A    (      s    i    )  −      F    A    (      s          i      −      1        )  ) , except the boundary terms, but that is fine, since   (      F    A    −      F    B    )  (      s    n    )  =  0, and   g  (      s    0    )  =  0, where       s    0   is some other number smaller than the smallest number in   S.The wikipedia page on summation by parts explains it as well I think.

Suppose that λ 1 </msub> ( A ) ≥

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